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Journal of Experimental and Theoretical Physics

Polarization effects in the interaction between light and multilevel quantum systems

K. A. Nasyrov, Anatolii M. Shalagin

Published Aug 1, 1999• 0 citations
Publisher

Abstract

We develop a theory that describes the interaction of radiation and multilevel particles on the basis of the classical description of the orientation of angular momentum. For examples of application of this theory we take the problems of stimulated Raman scattering and of parametric generation of light in four-wave mixing. We find that the efficiency of the radiation processes largely depends on the polarization of the pump waves and on the types of optical transitions (Δ J ) and to a much lesser extent on the value of the angular momentum J . We also explain some polarization phenomena observed in experiments on four-wave mixing.

DOI
10.1134/1.558975

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We develop a theory that describes the interaction of radiation and multilevel particles on the basis of the classical description of the orientation of angular momentum. For examples of application of this theory we take the problems of stimulated Raman scattering and of parametric generation of light in four-wave mixing. We find that the efficiency of the radiation processes largely depends on the polarization of the pump waves and on the types of optical transitions (Δ J ) and to a much lesser extent on the value of the angular momentum J . We also explain some polarization phenomena observed in experiments on four-wave mixing.
We generalize an approach (previously proposed by two of the authors) to describe quantum systems with large angular momentum exposed to a radiation field. The approach exploits a transformation from the quantum-mechanical description to the classical one with respect to angular momentum orientation. The transformation is carried out through a density matrix representation similar to the well known Wigner representation for the translational degrees of freedom. The generalization opens up the possibility of describing the interaction of polychromatic radiation (i.e. a set of monochromatic components) with a quantum system when the spectral components of the radiation are resonant to different transitions of the system. It follows naturally from the obtained equations that if the solution of the problem within the framework of the model of nondegenerate states is known, then its generalization for the case of state degeneracy and accounting for polarization of interacting fields is given by the simplest operation. As an example, the proposed method is applied to stimulated Raman scattering and four-wave mixing. We pay special attention to those polarization features which, on the one hand, are interesting to experimentalists and, on the other hand, have not yet been given an adequate explanation in the literature.
In recent years interest has significantly quickened in investigation of nonlinear optical and spectroscopic phenomena in such unique objects as Rydberg atoms (the atoms in the energy states near the ionization limit) and van der Waals complexes (dimers). These objects are characterized, in particular, by that their energy states are strongly degenerate due to a large value of angular (rotational) momentum. Here, the problem arises of adequate theoretical description of experimental results, especially when polarization peculiarities of nonlinear radiation processes are considered. The problem is caused by the existence of a huge number of quantum magnetic sublevels of energy states, which are, in the general case, coherently coupled to each other and take part in the radiation processes. Figure 1 gives a vivid presentation of the problem arising even in such a seemingly simple case when one need describe the radiation transitions caused by a strong monochromatic field between two energy levels with large values of the angular momentum J. If the field polarization is arbitrary, the whole set of the magnetic sublevels turn out to be coupled by radiative transitions. Formally, the problem becomes essentially multilevel. In this case not only the hope fades of solving it analytically using the traditional quantum-mechanical approach, but the problems of numerical solution grow abruptly with increasing quantum number J as well. On the other hand, as is often the case, every cloud has a silver lining. With large J we approach the situation when the rotational motion of particles can be described classically. It turns out that the transition from the quantum description of orientation states of the angular momentum to the classical one simplifies the problem radically: it reduces to the problem of radiation processes in the simplest model of nondegenerate states. In this case the rotational motion manifests itself in a dependence of the density matrix elements on orientation angles of the angular momentum as on a parameter.
The Wigner representation of angular-momentum orientation in the classical limit on $J\ensuremath{\gg}1$ has been used to derive an equation of angular-momentum motion in the case of a closed two-level system interacting with an arbitrary polarized radiation. This equation enables us to carry out a theoretical analysis of angular-momentum motion and consider final states of the quantum system for all types of dipole transitions. We have found that in the case of a zero magnetic field in the final state of the system with $\stackrel{\ensuremath{\rightarrow}}{J}J$ or $\stackrel{\ensuremath{\rightarrow}}{J}J+1$ optical transition, the angular momentum is directed along or opposite to the wave vector of the radiation depending on the sign of the polarization ellipsity, while for a linear polarization the angular momentum is isotropically distributed on the plane orthogonal to the polarization vector. For the $\stackrel{\ensuremath{\rightarrow}}{J}J\ensuremath{-}1$ transition two directions of angular momentum are possible in the final state. These directions are determined by the ellipsity of polarization. The spatial size of the final distribution of the angular momentum has been defined by the quantum uncertainty, 1/J,1/\sqrt{J},1/J​, of the angular-momentum orientation. In the case of the $\stackrel{\ensuremath{\rightarrow}}{J}J\ensuremath{-}1$ or $\stackrel{\ensuremath{\rightarrow}}{J}J,$ transition particles come into ``dark'' states while for the $\stackrel{\ensuremath{\rightarrow}}{J}J+1$ transition they occupy the ``brightest'' state. In the presence of a nonzero magnetic field, particles with $\stackrel{\ensuremath{\rightarrow}}{J}J$ transition have in the final state only one angular-momentum orientation, i.e., along or opposite to the direction of the magnetic field depending on the polarization ellipsity sign. For the $\stackrel{\ensuremath{\rightarrow}}{J}J+1$ transition both directions are possible, i.e., along and opposite to the magnetic field. In the case of the $\stackrel{\ensuremath{\rightarrow}}{J}J\ensuremath{-}1$ transition the directions of the angular momentum form a cone around the magnetic field.
A theory of resonantly enhanced degenerate four-wave mixing in two-level systems including the effects of atomic and photon angular momentum is presented in the absence of pump-induced saturation of the transition. It is shown that there exist three distinct quantum-mechanical amplitudes leading to the third-order polarization density. These quantum-mechanical amplitudes are shown to be sensitive to the states of polarization of the incident fields. The quantum-mechanical transport equation in the mmm representation is used to calculate the output signal in the various regimes of laser detuning, atomic linewidth, and polarization states of the radiation field for collinear interaction.
Stimulated Raman scattering of linearly polarized light from degenerate and selectively populated atomic sublevels of potassium have been observed. The scattered radiation has a polarization perpendicular to that of the incident beam. Such transition can take place between substates of different or of the same atomic level. In the latter case, the frequencies of the incident and scattered beams are equal and the polarizations are perpendicular. The induced transitions are associated with parametric four-wave coupling possessing also characteristic polarization properties.
Polarization effects in stimulated Raman and stimulated Rayleigh scattering and self focusing are investigated from the viewpoint of the scattering of polarized photons by molecules. Both stimulated vibrational Raman scattering in liquids and stimulated rotational Raman scattering in gases are considered, and it is demonstrated that the polarization changes and relative power thresholds for different laser polarizations should be somewhat different to those predicted in other theories. For example, the ellipticity eta 0 of the stimulated rotational Raman photons in the forward direction from Delta J=+or-2 transitions in gaseous symmetric top molecules is given in terms of the initial ellipticity eta 1 by sin 2 eta 0 = - 5 / 7 sin 2 eta 1 , whereas conventional theories predict that eta 0 and eta 1 are equal in magnitude and opposite in sign. Furthermore it is shown that deviations from this relation could provide a measure of the degree of optical orientation of the molecules.